Financial & Strategic Value Creation 103: Power of Compounding & Discounting
Disclaimer: The views expressed here are personal and meant for information/education purposes only and not representative of his employer directly or indirectly. Any mention of company names are for illustrative examples only.
This article is part of a series on Financial & Strategic Value Creation written for Technologists / techies: "Financial & Strategic Value Creation for Techies 101", "Financial & Strategic Value Creation for Techies 102: Analysis & Valuation", "Financial & Strategic Value Creation for Techies 103: Power of Compounding & Discounting", "Financial Literacy 104: Want Growth?", "Financial Literacy 105: Strategy Meets Execution"
PS: Selected older articles (2016-18): Blockchain/Crypto/Digital Tokens, Cashless Finance: UPI, Blockchain/Agri & Supply Chain Finance, Digital Agri: Commodity Spot / Futures Markets, Digital Agri: Future of Insurance, From Barter to Blockchain: Brief Journey of Payments & Ledgers, Cashless Financial Inclusion...
Albert Einstein is supposed to have said that compound interest is the eighth wonder of the world or the most powerful force in the universe. The power of compounding of growth is also the most important thing to know about finance: if you forget everything else in finance, remember some of this. Discounting (as in "discounted cash flows" (DCF)) is the inverse of compounding, and the value of a company is simply the sum of discounted cash flows. Think of compounding as measuring impact of "growth" & "efficiency" and discounting as "risk" covered in prior articles.
In this article we explore some practical facts as questions that business people tend to use / remember in their day to day mental calculations. The appendices give more details. First the preliminaries.
Preliminaries: Compounding & Discounting Definition, Discounted Cash Flows (DCF)
Compounding: P is the principal, g is the annual interest rate (CAGR, Compound Annual Growth Rate), and n is the number of years of compounding, and F is the amount accumulated post compounding, aka Future Value (FV).
In the "continuous compounding" model (derived later), the above discrete formula becomes the simple exponential formula P*e^gT where g is the growth rate (eg: 0.25, or 25%, limited by return on incremental capital) and T is the continuous time period (equivalent to n discrete intervals earlier).
In other words, compounding leads to an "exponential" increase in capital. Compounding with higher growth rates (or return on capital) for a longer period of time T leads to tremendous wealth even if the initial principal P invested is lower. Remember that "exponential" increase at high growth rates, sustained for long periods is what leads to huge wealth. Doubling in 3 years (26% CAGR) for 10 times means a 1000X increase in wealth in 30 years (or 10X in 10 years (~3*3.3); 100X in 20 years (~3*6.6); 1000X in 30 years). The picture above shows 15% CAGR where $1 becomes $2 (i.e. doubling) in 5 years, $4 in 10 years (second doubling) and $15 in 15 years... i.e. 15% CAGR corresponds to doubling capital every 5 years. Note that the capital invested is 1, but the interest accumulated (ignoring taxes etc) is 7X more.. To get to 1000X at 15% growth rate you need 50 years (5*10) rather than 30 years with 25% doubling. You can retire 20 years earlier, since you have made your money work for you!
Discounting: You divide by (1 + r)^n in discounting to get a "Present Value" (PV) of a future number (eg: or a sequence of cash flows) whereas and you multiply in compounding to get the "future value" of current principal. This is the "discrete" model for discounting.
In a "continuous discounting" model (vs a discrete model above), the discounting effectively is an exponential function with a negative exponent -r (corresponding to the discount rate r), and T with time period (equivalent to n intervals in the discrete case). This is why the discounting curve looks like a negative exponential, because that is what it is!! A depreciating asset will lose money exponentially at the rate of discounting or depreciation!
The discount rate a.k.a "cost of capital" is a function of "risk" (more risky, higher discount rate, r due to "risk premiums" and steeper "discount" on future cash flows). Think of r as a price (or penalty) of risk that increases in intensity with time. It is usually a function of three components: a risk-free rate (roughly that tracks inflation rates long term, and sometimes an added component based upon central bank policies to combat inflation or deflation), a term premium (longer term loans in general cost more than shorter term (except for short phases when the yield curve is inverted)), and a risk premium (either a default spread for bonds, equity risk premium for equities, or currency risk premium that captures long term differences in inflation if you are also converting currencies in a valuation).
Exponential declines means that cash flows far out in the future are exponentially smaller in net present value (NPV) terms. Therefore, any reduction in operating or financial risk will lead to huge increases in present value. The exponential impact of discount rate is why reduction of risk is as important as improvement in the absolute magnitude or growth of cash flows. The picture above shows the impact of 15% discount rate, i.e. halving every 5 years (the inverse of compounding): $1 becomes $0.5 in 5 years; 0.25 in 10 years and 0.125 in 15 years. Cash flows far into the future are significantly discounted.
[Learning Curves in LiIon and EVs:] One of the well known phenomena in manufacturing companies is the so-called "learning curve" where the cost-per-unit drops by certain % with the cumulative doubling in number of units produced. The sources of learning could be multi-faceted (better system integration and packaging innovations, lower labor or materials cost per unit, or in the context of AI-based learning (eg: for autonomous driving) the doubling or 10X growth in data available for AI algorithms to learn from). A discussion of "Wright's law" by ARK Invest is here. For Li-Ion batteries in vehicles and cost of electric vehicles, it is estimated that the learning rate is around 15-18% per doubling of production. So for every 4 or 5 doubling of productions for 18.8% and 15% drop respectively, the cost should halve (based upon the inverse-of-compounding rule). Now if every doubling happens in 1.5 years (18 months), 4 doublings happens in 6 years; and 5 doublings in 7.5 years. We should expect the cost of Li-Ion batteries to halve in 6 years from where they are today (irrespective of other metrics like energy density etc!). If Li-Ion battery packs are 40% of the cost of the car at around $15-16K today per pack, it will drop to ~$8K/pack by 2025. Game over for ICE vehicles by then, given the other metrics will also improve.
[Multiple Cash Flows: Compounding & Discounting:] We saw compounding / discounting of a single lump sum either forward or backward in time. When you have multiple cash flows, you just sum up the compounded / discounted value for each component cash flow for the appropriate period of time and associated rate of growth or discount factor.
When you have a tug-of-war or "tussle" between growth in the future and risk, the net will turn out to be more when growth outweighs risk, i.e. the "return on invested capital" (which bounds the sustainable growth rate g without additional capital) is greater than the (weighted average) cost of capital (r). If you need to grow at a rate faster than return on invested capital, you need to invest additional capital early on (this is where growth investors, secondary offerings or venture capitalists come in).
The growth-risk tussle or balance is elegantly captured in the continuous compounding/discounting framework where the exponent (g-r) shows how the growth net of opportunity cost during period of competitive advantage happens. The economic value added (EVA) happens when g > r and future value grows exponentially with the gap between g and r. Conversely, value destruction happens when g < r (i.e. when EVA is negative) and future value declines exponentially! The latter if sustained is called poor capital allocation.
[Discounted Cash Flow (DCF): Perpetuity & Annuity & P/E:] Assume you have a company (or government) that takes in a principal, and pays a stream of cash flows C at every year (coupons, dividends, interest, earnings, free cash flow etc) forever. This is called a "perpetuity". How much is the value of the cash flow? In other words if you get a lump sum lottery payoff and invest in this perpetuity, you are swapping your lottery windfall for a future stream of cash flows for several generations! Since this is a risk free stream of cash flows, you discount it with a risk free rate, r. The perpetuity formula: P = C / r , super simple!
If you think of C as the "earnings" then the "P/E" or price earnings ratio of this perpetuity is P/C = 1/r. It is useful to mentally note that in markets like US or Europe with very low inflation rates (eg: 2%), and small term premiums (0.5% or 50 basis points), the risk free rate is trading around 2.5%. This means that the P/E for "bonds" is 40 or sometimes more. If a stock is priced at a P/E of 15, then it implies a equivalent discount rate of 6.67%, or an equity risk premium of 6.67-2.5 = 4.17% of stocks compared to bonds. Equity risk premiums historically tend to be between 4%-6% (its around 5.5% at the time of the writing), and lower values (eg: <3%) are associated with "overvaluation" or "bubbles" ("expensive" stocks) in the market or in subsectors. Today it appears that the bond market is quite richly valued in US, and Europe (with negative yields in some markets to combat deflation!). And risk of increase in yields means lower prices.
A simple variant on the perpetuity is the growing perpetuity where the payments grow with time at a rate g. In this, the formula is also very simple & elegant: P = C/(r -g).
A few technical points. < r. First, note that long term growth rate (g) in a perpetuity must be bounded by long term inflation or risk-free rates (r), i.e. g < r, else the company will grow bigger than the economy (an impossibility), thanks to the magic of compounding! Second, note that C is the coupon or earnings at the end of the first period. If earnings today (or trailing twelve months, TTM) is C0, then C1 = C0*(1+g), which is also called the "forward earnings". Typically the discount rate r used here will also include the term premium and risk premium (eg: equity risk premium or default premium). If you flip it around and find a stock which is priced P. It is paying earnings E0, trading at normalized price P/E = P/E0, which implies a earnings yield of E0/P, or E0*(1+g)/P if you use the formula w/ forward earnings.
Annuity is just a truncated form of perpetuity, where the annuity is paid out for a finite number of periods N. A growing annuity with growth rate g is again a truncated growing perpetuity. The formulas for annuity and growing annuity are simply the perpetuity formulas C/r or C/(r-g), but scaled with a factor [1 - (1+g/1+r)^t] that has growth rate (g) on numerator and discount rate (r) in denominator, similar to the "tussle" we discussed earlier.
[DCF Valuation: Annuity + Perpetuity:] Companies cash flows are in broadly two parts:
- A "Competitive Advantage Period" (CAP) of higher growth for say 5-7 years {i.e. a growth annuity}, i.e. a C/(g1-r1) type term. The terms like "competitive advantage", "moat" etc factor into this, and
- A Terminal Value that corresponds to a growth perpetuity with a growth rate < risk free rate. The terminal value has to be discounted back to present value. C/(r-g2) * 1/r^n, type term for the discounted terminal value.
- Cost of capital or discount rate would depend upon the riskiness of the company. Prof. Damodaran tends to use different discount rates in each year, as the company transitions from a high growth, high risk to a more stable growth terminal value.
A DCF is actually simple: a C/(g1-r1) type term (with a scaling for growth annuity) plus a C/(r-g2) * 1/r^n, type term for the terminal value discounted. n is the number of years of competitive advantage. g1 is the growth rate in high growth phase; and g2 is terminal growth rate.
Discounted cash flow (DCF) valuation is just the summation of such discounted values to get to a "net present value" (NPV) or valuation of a project or company. Structurally, it can be approximated as the sum of two parts: a growth annuity (competitive advantage period) and a discounted terminal value. If a firm is expected to last for many years, and has a long competitive advantage period with solid growth, the terminal value term drives a lot of the value of the firm even though it is way into the future. Peter Thiel explores this in his talk (video, 27 min), and graph reproduced below.
[Equated Monthly Instalments (EMI):] You can also play with (i.e. invert) the annuity formulas to invert it get the value of C from P, or the future value F. There are corresponding excel formulas PMT to calculate EMI, and its components Principal Repaid (PPMT) and Interest Paid (IPMT). You can built out a loan amortization schedule with it for loans and monthly payment instalments which combine both interest and capital repayments. Some useful numbers to keep in mind (money lenders know this well!):
- Principal (Loan) P, $1000, Monthly rate 1%, 2 years (24 mo), EMI: $47.07 (per $1000 loan)
- Principal (Loan) P, $1000, Monthly rate 1%, 3 years (36 mo), EMI: $33.21 (per $1000 loan). Lower EMI, but higher total interest paid (~$191 vs $130)
- Principal (Loan) P, $1000, Monthly rate 2%, 2 years (24 mo), EMI: $52.87 (per $1000 loan). Note the EMI amount is not much larger than 1% loan (only 12% more), but total interest paid over 2 years has jumped from $130 to $268 (> doubled!)! Watch out when money lenders bump up EMI slightly!
- Principal (Loan) P, $1000, Monthly rate 2%, 3 years (36 mo), EMI: $39.23 (per $1000 loan). Again this looks lower than $47 and only slightly higher than $33, but total interest paid in 36 months has ballooned to $404!!
We will see monthly interest rates vs APR (annual CAGR) and continuous compounding later in the article also.
[Investment Expectations, Paybacks] To double (2X) in N years, what is the compound annual interest rate (CAGR) ? [CAGR %, Years-to-Double]
Business people (and good money lenders!) keep these quick associations in their heads:
- ~5% (annual interest rate) means doubling in ~14 years [5%/y, 14.2 yrs]
- ?~7.5% means doubling in ~10 years [7.5%/y, 9.6 yrs]
- ~10% means doubling in ~7 years [10%/y, 7.25 yrs]
- ~15% means doubling in 5 years [15%/y, 5 yrs], ...
- ~20% means doubling in ~4 years [20%/y, 3.8 yrs]
- ~25% means doubling in 3 years [25%/y, 3 yrs] ...
- ~40% means doubling in 2 years [40%/y, 2 yrs]
- ~60% means doubling in 1.5 years (18 months) [60%/y, 1.5 yrs]
- ... and of course 100% doubles in a year. [100%/y, 1 yrs]
Doubling in 5 years (15%) vs Doubling in 3 years (26%) may not look impressive enough. What is 3 years vs 5 years -- only a two years difference! But the true power of compounding occurs when you repeat this process, say 10 times. Over 30 years, doubling in 3 years leads to > 1000X ($1 million becomes $1 billion, yes with a B... or for Indians, Rs. 10 lakh becomes Rs. 100 crore!). This is where you remember Einstein. It is the COMBINATION of higher g (return on capital invested into growth or "quality premium") AND longer compounding period (of sustained growth compounding) that delivers the magical outcome of the exponential.
[Investment Returns: Stock markets & ROCE/Business hurdle rates:] The [15%/y, 5 yrs] and [25%, 3yrs] are common in India in business investment decisions as hurdle rates for internal financial / capital allocation decisions. From a fundamental valuation perspective, strong companies like HDFC, Asian Paints etc have ROCE (return on capital employed) in the 20-25% / year range (which means doubling equity base in 3-4 years). Long term stock returns tend to mirror long term ROCE or similar return on (equity) capital metrics. The average return expectation in India given risk free rate of ~7% and e
quity risk premiums of 6-8% has meant a expected return of 13-15% on the stock market which translates into expectations of doubling in ~5 years in rupee terms for the index. It also turns out that the CAGR of the Sensex over 30 years is approximately 14-15%.
The number 15% is therefore a good benchmark for ROCE (return on capital) for the long run to be above cost of capital in India [this view is also taken by the Coffee Can Investing Approach by Saurabh Mukherjea] at this time of writing. Quality stocks are expected to double (considering both dividend and capital appreciation) in 3 years vs the index in 5 years. In the US equity market (eg: S&P 500), with a risk free rate of 2.5% and equity risk premium of 5-5.5%, it translates to a expected return of 7.5-8% or a capital doubling in dollar terms of 10 years. If the rupee returns on Sensex are significantly higher than dollar returns in S&P 500, why doesnt money rush to India? Partly, the explanation attributed to currency risk (dollar vs rupee), difference in country risk (India vs US), inflation risk differentials (4-5% vs 1-2%). As currency risk, country risk and inflation risks subside in India, while real growth remains strong, there will be more capital allocated since real returns are higher.
More precisely, the math says:
- Doubling in 1 year => 100% annual rate
- Doubling in 2 years => 41.45% annual rate (approx 40%) or 2.89% monthly rate
- Doubling in 3 years => 25.98% interest rate (approx 26%) or 1.92% monthly rate
- Doubling in 4 years => 18.88% interest rate (approx 19%) or 1.43% monthly rate
- Doubling in 5 years => 14.83% interest rate (approx 15%) or 1.14% monthly rate
- Doubling in 6 years => 12.18% interest rate (approx 12%) or 1% monthly rate
A 10% CAGR (annual interest rate) will double in ~7.27 years; and 5% CAGR will double in 14.2 years. And 10 years to double will need ~7.5% CAGR.
A quick approximation is CAGR r = 72/N for practical numbers. Eg: 72/3 = 24% ... 72/5 = 14.4%, 72/2 = 36%. This tends to understand the CAGR a bit for 2-10 years (if you compare the numbers above).
The inverse of compounding is discounting. At a 26% discount rate, $100 three years in the future is worth $50 today, and for a 15% discount rate, $100 five years in the future is worth half the amount today. Similarly at 10% discount rate, $100 seven+ years away is worth $50 today.
The day-to-day short term fluctuations are less important than the fundamental compounding rate of the cash flow engine. This is why if you find a long term compounding stock, it is better to ride the ups and downs of the market or add some more principal during the dips. Remember in the formula F = P * e^rt the P (principal) term is a "linear effect", whereas the compounded growth e^rt term is the dominant term with "exponential effect" on the long run if r (compounding engine) is strong, and you give enough time t.
[Long-term Investing in high P/E, high growth, high ROC stocks:] Even if a excellent company with high ROCE & cash flow growth is overvalued in the short term (3-5 years), the compounding of cash flows will catch up to its valuations. Holding periods of 10+ years will make timing less relevant for a long term investor or value-creating manager.
Look at the graph above. If the company sustained return on capital is g1 (recall that growth rate is limited by return on capital and reinvestment %), and market growth rate is g2. The P/E of this high quality, high secular growth stock (eg: Google, Facebook during its prime growth years) is significantly higher than a market benchmark, lets say even 2X i.e. double the P/E. What this means is that if you buy a normalized P unit of the market benchmark, you can only buy P/2 of the high quality high growth stock. After a long period of time T (say 10-15 years), the market investment becomes F1 = P*e^g1T and the stock investment becomes F2 = P/2 * e^g2T.
Even though you have invested only half normalized units in this "expensive" stock, the power of compounding means that the e^g2T >> e^g1T which means that the expensive stock will still outperform the relatively "cheap" market. The required P/E to ensure similar returns as the market is significantly higher (sometimes > 100 etc) since it has to combat the exponential term to equalize the net returns. The effect of high P/E is only affecting the linear term (initial principal invested, appropriated normalized). The final value of F1 becomes double of F2 in about 15 years, even though you bought it at double the P/E of the market! Saurabh Mukherjea in his book on "Coffee Can Investing" says that even if the P/E of the stock declines with time towards the market P/E, the high years of initial growth will lead to significant alpha or excess returns.
Michael Maubossin from Credit Suisse in a famous paper on P/E multiples has the following exhibit. Cost of capital (WACC) is assumed to be 8% here and the "steady state" or commodity P/E is 1/8% = 12.5.
For companies who destroy value, i.e. their ROIC < 8%, they have P/E < 12.5. In fact with 4% ROIC and 6% growth or higher, the P/E drops to 3.3 or lower: these companies need a lot of capital to keep growing which they are investing without earning their cost of capital so the market withdraws capital & raises the price of capital (i.e. marks down the price). P/E of 4 implies a yield expectation of 25% (vs a nominal of 8%)! Similarly for a EVA positive company of 16% or 24% they can command a higher P/E. The incremental P/E is a function of the spread of ROIC and Cost of capital, invested capital for the desired level of growth and expected period of such value-accretive growth (also called the competitive advantage period / CAP). For details read the paper!
[Value Investing in Growth Stocks with High ROCE:] Can we model the target P/E given the ROC (v), Cost of Capital (r) and Growth rate (g)? Here is a first approximation for the numbers above. The principle is simple: When you were earning return on capital (v) equal to cost of capital (r), then the P/E is 1/r (assuming a simple perpetuity model), and 100% of earnings is retained and compounded at rate r. When v is different from r, then a fraction g/v of earnings is retained and compounded at rate v (retained earnings or retention ratio), and the remaining earnings fraction (1 - g/v) is paid out (dividends or payout ratio) which the shareholders can get a return equal to cost of capital of r.
The weighted average shareholder return is now: w = (1 - g/v)*r + (g/v)*v. If you simplify this, w = (g + r) - (gr/v). What this formula says if the company has ROC (v) > Cost of capital (r) AND it has growth opportunities for the incremental capital of g (ideally equal to v), then it can preferably reinvest the earnings and earn an "excess (economic) return" (v - r) or "economic value added" (EVA) for its shareholders. If the old P/E corresponded to a return of r is 1/r, the new P/E corresponding to weighted average return of w is just the scaled up P/E: (1/r) * (w / r), which simplifies to P/E = {v(g+r) - gr}v.r^2 = {g/r^2 + 1/r - g/vr}.
Despite the difference in forecast periods and using a simple scaled perpetuity model, this formula is a good first order match in the estimate of New P/E using the above formula (see below). Why is this useful? When you are evaluating a company with apparently high P/E, you can plug in these numbers ROCE (v), earnings or FCFE growth (g) and rough estimate of Cost of Capital (r) to get a first cut estimate of what its P/E should be, and see if you are getting value or not. Note the formula tends to underestimate target P/E (i.e. be conservative) at higher growth rate (g) and higher ROCE (v): if a high ROCE, high growth stock is under this predicted P/E, then it is worth a closer look.
This myopia about P/E without understanding the drivers is why many investors missed or did not invest in a sustained period in great ROCE stocks like Google, Amazon etc.
Lesson: Don't miss an Amazon or Google because P/E appeared to be high. If there is secular growth accompanied by excellent capital efficiency, the P/E can be much higher than market and still give excess returns.
[Growth or Productivity/Efficiency (ROE): What to Focus on?:] An excellent insight from Mr. Raamdeo Agrawal in his 2018 Wealth creation study was that companies with low(er) ROE than cost of capital should focus on increasing ROE and return on incremental capital; and companies with high ROE and low growth can get a larger marginal impact on value by raising growth rate.
Typically companies have a period of such super-normal growth (or period of competitive advantage), and then they have a long term growth rate that matches the sector / economic growth rate or slightly higher. Else they will become bigger than the economy itself (and become significant drivers in the overall economic productivity of the nation. As the tech giants like Amazon, Apple, Microsoft, Google, Facebook approach or cross trillion dollar valuations and have accounted for most of the S&P 500 growth in recent years, they are starting to collide with political & anti-trust constraints across the world since they are all pervasive in the economy.
[Payback / IRR]: Business project justification frames a decision sometimes as "payback" period (or "cash-back" or principal recovery period), roughly how quickly you want to get your principal capital back (within an implicit assumption of asset not significantly depreciated). This is also a statement about doubling implicitly. Payback in 3 years roughly implies 25-26% IRR (internal rate of return) over three years, assuming the asset has not depreciated and principal value still holds. Payback in 2 years implies 42% IRR, and 18 month (or 1.5 year) payback implies 60% IRR or hurdle rates during such payback periods! Sometimes such punitive hurdle rate / cash-back expectations could kill interesting projects due to high implied IRRs. On the other hand when a business is focused on cash, the goal may be to recover investments and get back cash on the short run.{note that IRR is usually calculated over the entire cash flow period, but we have used it in the context of the payback period, and assumed no depreciation of principal.} IRR is a form of return or CAGR or "interest rate".
The impact of even an apparently small differential in CAGR ("excess returns") can exponentially compound into a very high. Consider 10% CAGR vs 15% CAGR. By year 7.25, $1 doubles in the 10% CAGR case to $2. However, we would have achieved around $2.8 (i.e. 40% more aggregate capital accumulation) in the same period with 15% CAGR. This difference continues to compound further. By year 15, we end up with $4.18 (10% CAGR) vs $8.14 (15% CAGR) or a ~100% relative differential in capital, and a 400% differential on the base of $1 we started with. The gaps widen with 20% CAGR to $15.41, i.e. another ~100% relative differential vs 15% CAGR case, i.e. double the absolute gap.
[Fund or AUM Expense Ratios:] Transactions costs and fund expense ratios or AUM (asset under management) expense ratios in markets like the US have come down significantly over the years. This is however an issue in emerging markets like India. Sourabh Mukherjea in his book Coffee Can Investing gives an example of a 15% gross return portfolio, with two funds: one with expense ratio of 2.5%-per-year (avg equity mutual fund) and another with 0.1%-per-year (eg: an ETF). The impact of this 240 basis points per year cost on a portfolio of Rs. 1 lakh is that the second fund's corpus exceeds the first by 24% after 10 years; 53% after 20 years; 89% after 30 years and 133% after forty years. At forty years the two corpuses are Rs. 1.1 crore vs Rs. 2.6 crore!
Lesson: don't let compounding work against you; and small differences compounded over long periods matter!
We shall see later than when we do "continuous" compounding, then the compound interest formula is a true exponential, i.e. A = P*e^rt. One property of the exponential functions is that the derivative (or rate of change) of e^x is the same function, e^x !
This is simpler to understand in 2^x, where the difference between 2^3 to 2^4 is of the same magnitude as 2^3 (i.e. the difference itself is an exponential function 2^x, and not a linear function of x).
The true power of "exponential" compounding or growth, comes with "sustained compounding" i.e. the emphasis on sustained period or length of compounding (number of periods of doubling). This is why Einstein called this the eighth wonder of the world, and why buy-and-hold or adding-to-dips or systematic-investment-plans (SIP), aka dollar-cost-averaging into great compounding machines may be superior to trying to time getting in / out.
Qn 2: [Corpus Growth, Segment CAGR] What does 5X in 10 years mean in % CAGR?
This is a 1000X growth of capital. This is the power of compounding shown below:
- Step 1: Pick investments such that capital doubles in 3 years (~26% annual return)
- Step 2: 10 doublings (i.e. 3*10 = 30 years) means 1000X growth of capital (2^10 = 1024).
Now you can understand how an investor like Warren Buffett who returns ~25% over a VERY long time (30 years+) can lead to 1000 times growth in your money invested w/ him. Or 3+ doublings @ 25-26% means 10X growth (2^3 = 8) in about 9-10 years.
Assume that alternate investments yielded only 15% (or doublings every 5 years). This means six doublings or 2^6 = 64X. While 64X looks impressive relative to 1X, it pales in comparison to 1000X which is 16 times larger accumulated capital in relative terms and simply huge in absolute terms.
[Market Segment CAGR] See the graph below showing 15% CAGR and then a sharp spike up showing 5X growth in next 10 years [SME finance requirement]. Is the latter growth really such a large spike up from 15%? Lets see!
We saw that 10X in 10 years is a little more than 3 doublings in 3 years compounded. 5X in 10 years is approximately 2*5 a bit more than two doublings every 5 years which will give 4X in 10 years. Since it takes 15% to double in 5 years, it turns out bumping it up to 17.5% CAGR will give you 5X in 10 years. 17.5% is meaningfully higher (250 basis points). But even this when compounded over a 10 year period, and applied to a higher base (600 in the picture) leads to a significantly higher end point. Also from a "presentation" standpoint sounds more impressive to say 5X in 10 years vs just 17.5% vs 15%, accompanied by the significant area graphed!
The graph also shows that the base principal also matters in combination w/ compounding rate and the period of compounding. When Bill Gates retired he was the richest person in the world. He diversified some of his wealth from Microsoft, but held a significant amount of shares. When Satya Nadella took over as CEO of Microsoft, the stock took off again, and Gates' higher principal combined with higher CAGR under Nadella led to huge accumulation of capital for his philanthropic activities. Jeff Bezos however held onto a bigger compounding machine (faster CAGR of Amazon stock) and overtook Gates as #1.
The appendix gives more details for other numbers (eg: 3X in 3 years or 10X in 10 years). In general, approximate the number (eg: 7X) with the nearest power of 2. Eg: for 7X, use 8 = 2^3 ... In other words, for 7X in 7 years you can guess that you need a bit less than ~three doublings (2^3), which means you need bit more than a doubling every 7/3 = 2.3 years. This is approximately 33% CAGR.
[VC Valuation: 10X in 5 years if successful. ~Zero if Not! What is Blended IRR?:] When VCs value companies, they expect dilution of stake, and significant return if successful exit. A rule of thumb is 5X in 10 years (eg: see this video). A quick check yields that this is 60% CAGR over 5 years. In this case success CAGR is not the same as IRR, since the probability of success is only 5-10%: there is a higher probability of failure where the outcome is zero or possibly a weak exit at 10% CAGR on an averaged basis across holdings. So a blended IRR assuming a 10% success rate would be 60%*0.1 + 10%*0.9 = 6 + 9 = 15% IRR which is similar to stock returns in Sensex. Also since only a few positions in a VC portfolio (10-20%) turn out to be winners, the goal is to maximize the upside on winners. Winners are bearing the costs of the losers!
Qn 3: [Credit Cards, Microfinance, MAU for Startups] What is 3% monthly interest rate in CAGR (annual interest rate) & Doubling-Years?
Naively we may think 1% per month will mean 12% per year, but no, it is more! (almost 13%/ year or 90 basis points higher than simple calculation!). This effect is magnified for larger monthly interest rates. 2% per month is not 24%, but 27.2% (320 basis points higher!). 3% per month is not 36% but corresponds to 43.2% CAGR (720 basis points higher). This is the power of compounding. And it can play against you if you do not watch out!
More precisely:
- 1%/month means doubling in ~6 years [1%/mo, 6 yrs, 12.9% CAGR];
- 2% / mo => ~3 years [2%/mo, 3 yrs, 27.2% CAGR];
- 3% / mo => <2 years [3%/mo, 2 yrs, 43.24% CAGR]
Credit cards often charge 2-3% per month as interest rates. You may get lulled into thinking 2-3% is a small number, but at 27.2%, the bank is doubling its money loaned to you in less than 3 years, and at 3%/month, you are helping the bank double its money in < 2 years (1.92 years to be precise!). You want to be on the other side of this trade, i.e. the lender, and not the borrower here ! Do you know who else lends at 3% per month? Money lenders in villages who capture farmers in a debt trap (see below)! This is why it is better to pay off / consolidate / refinance unsecured credit card debt into a lower cost finance vehicle (eg: with some collateral or based upon a banking relationship home equity loan / loan-against-property) and pay off credit card loans first. Even personal unsecured loan rates in India are around 15-18% from banks (doubling in 4-5 years) where there is an existing banking relationship.
Microfinance/Money Lenders/Debt Trap: A lot of finance in emerging market is on an informal basis via money lenders, pawn brokers etc who may use gold etc as collateral/security, or lend to individuals and SMEs higher rates (3%-per-month) without security. Micro-finance companies come in and propose reduction from 3% to 2% a month, i.e. from 43%+ to 27% interest rates. This is a HUGE improvement to help poor folks refinance their costlier debt with relatively lower cost loans. However, in absolute terms compared to the higher layers of the pyramid these are very high rates.
These interest rates can be also viewed as cost of capital. Since poor folks do not pay any taxes, they do not get any tax advantage of debt and costs as much as equity capital. For companies, the cost of debt is D*(1 - tax rate), but as the corporate tax rate for poor folks is zero and for others is lower, the effective cost of debt capital goes up. If the cost of debt is 43%+, this becomes the hurdle rate for their economic activity. In other words if they cannot convert this capital into at least 43% of operating earnings (EBIT), all their operating earnings will go just to pay off debt. If the usual prudential benchmark of 2X interest coverage ratio is applied, then it implies that the return on capital has to be double, i.e. 86%!
Very few enterprises have such returns on capital, therefore, very high rates of interest also means that this covers only very small fraction of capital needs (eg: minimal / short term working capital). But unfortunately the more common outcome is such usurious rates of interest lead to a debt trap where the earnings fall short of interest coverage; and the farmer cannot repay the loan. However to avoid defaulting the farmer takes a second loan to pay off the first, but still does not have the income to service the second one since the second one is also a bigger loan (to cover interest and principal) and has the same 3%-per-month type super-high interest rate. The only solution is a combination of default / bankruptcy and a fresh start with a lower rate of interest on loans.
[Growth of Monthly/Weekly/Daily Active Users:] Many online startups today accumulate users or traffic first prior to "monetizing" them to get to revenue. The value of a user involves a cost to accumulate and retain a customer; and the lifetime value of a customer. Such startups measure the growth of their traffic or user base on a periodic basis (monthly/weekly/daily) and focus on the active user set. They use metrics like compounded X growth rate, where X = monthly, weekly or daily. This corresponds to the formula we shall study shortly below.
A good overview is here and here. Daily active users (DAU) is the total number of users that engage in some way with a web or mobile product on a given day. In most cases, to be considered “active,” users simply have to view or open the product. Web and mobile app businesses typically consider DAU as their primary measure of growth or engagement.
Monthly active users (MAU) is the aggregate sum of daily active users over a period of one month. In most cases, to be considered a “monthly active user,” a person has to open or view an app at least once in the period of one month. The ratio of DAU/MAU is typically a measure of ‘stickiness’ for internet products.
Given the numbers we have seen earlier, a 1% per month (12.9% annual) or 3% per month MAU growth (43+% per year) is very different. Fast growing companies may have even higher MAU growth numbers like 5%-per-month in their earlier stages of growth.
Compounding Periods: Annual, Half-Yearly, Quarterly, Monthly, Daily & Continuous
It is better to compound more often. Consider a 12% nominal annual interest rate. If you compound once in a year, you get one interest payment which is 12% of the principal at the start of the year. If you compound monthly, the interest earned in a specific month will also get compounded in subsequent months. The power of this interest on interest can be significant. At the limit, it is called "continuous compounding" and is also the source of the magical number, Euler's constant e = 2.71828 (for excellent videos, see here and here and here).
Exponential functions are proportional to their own derivative, but for the special constant e as the base, the derivative of e^x is itself, i.e. e^x. For example at x = 1, the slope (or loosely the change in capital at that point time) is e^1, i.e. equal to the amount of capital at that point in time. As capital doubles, the marginal increase in capital (i.e. the slope) also doubles. This turbo charges your increase.
Where does e come in? Consider doubling of your money in 1 year, i.e. $1 growing to $2. This is also equal to a nominal 100% return. 2^x = (1 + 100%)^x. If you compound twice a year (half yearly), $1 grows to $2.25, which is (1 + 100%/2 ) ^2. If you do this quarterly, you get $2.441 which is (1 + 25%)^4 (see below for details). For compounding n time periods, you get (1 + 1/n)^n ... The more the number of time periods, the compounded value increases from 2 successively but reaches a limit which is the irrational number e = 2.718
Monthly compounding: If you divide a nominal annual rate R by m periods (eg: m = 12 months for a monthly rate), and compound interest computed each of the m-periods, what is the effective CAGR or APR (annual percentage rate) ? For example R = 12% and m = 12 months, would give a 1%-per-month interest rate. What is the annual APR? It is not 12%, but higher 12.9% !
Coming back to the 100% compounding example, you could try weekly compounding (52 times a year) or daily compounding 365 times a year. The picture shows the convergence from 2 to e = 2.718 as the number of compounding instances goes from 1 to n. To a very good first approximation weekly or daily compounding is close to continuous compounding. In other words, you can get 35-36% more relative returns by continuous compounding rather than one interest payment at the end of the year!
[Trading] The following spreadsheet shows that even small but consistently positive daily or weekly compounding can lead to tremendous compound annual returns. Traders (whether in stock markets or real markets or market makers or robo-trading algorithms) who make small but consistent spreads and attempt to minimize or eliminate risk can make a large fortune with high returns. Eg: an average 0.2%-per-day gain compounded for 365 days gives a 73% return; or a 1%-a-week gain for 52 weeks gives a 55% annual return. The challenge is sustaining these returns as your corpus grows since the strategies will not work any longer or less effectively at scale. The take away message is that the reduction of risk, and consistency of compounding steady, but small gains can be a productive strategy.
This article is part of a series on Financial & Strategic Value Creation written for Technologists / techies. The earlier articles are here (102) and here (101).
Appendix:
P = Principal, r = Annual rate of interest (aka APR), N = # of years
- Compounding: F = P*(1+g)^n
- Discounting: P = F/(1+r)^n ... time-value / opportunity cost of capital and risk.
- Doubling Question: 2P = P(1+r)^N ... What is N ?
- Years-to-Doubling N = ln (2) / ln (1+r) which is also approximately equal to 72/r
- Multi-period compounding: r = (1 + R/m)^m - 1
- As m -> infinity, the continuous compounding/discounting formula converges to e^rt
Table 1 [CAGR, Years-to-Double]
- 5% CAGR => 14.2 years (~14 years)
- 7.5% CAGR => 9.6 years (~10 years)
- 10% CAGR => 7.27 years (~7 years)
- 15% CAGR => 4.95 years (~5 years)
- 20% CAGR => 3.8 years (<4 years... 18.88% = 4 year-doubling)
- 25% CAGR => 3.1 years (~3 years ... 26% = 3 year-doubling)
- 30% CAGR => 2.64 years (~2.5 years)
- 35% CAGR => 2.3 years
- 40% CAGR => 2.06 years (~2 years ... 41.47% = 2 year-doubling)
- 50% CAGR => 1.7 years
- 60% CAGR => 1.47 years (~1.5 years)
Table 2 [CAGR, NX in N years]
What CAGR does 5X in 5 years mean? More specifically NX in N years:
- 2X in 2 years => ~41.47% CAGR
- 3X in 3 years => ~45% CAGR
- 4X in 4 years => ~41.47% CAGR {same as 2X in 2 years}
- 5X in 5 years => ~38% CAGR
- 6X in 6 years => ~34.85% CAGR (~35%)
- 7X in 7 years => ~32.1% CAGR (~32%)
- 8X in 8 years => ~29.8% CAGR (~30%). Note it is NOT the same as 4X in 4 !
- 9X in 9 years => ~27.68% CAGR (~28%)
- 10X in 10 years => ~27.68% CAGR (~28%)
Observe that the required CAGR is declining from around 45% to 30% in steps of 300 basis points (3%) and later 2%.
This article is part of a series on Financial & Strategic Value Creation written for Technologists / techies: "Financial & Strategic Value Creation for Techies 101", "Financial & Strategic Value Creation for Techies 102: Analysis & Valuation", "Financial & Strategic Value Creation for Techies 103: Power of Compounding & Discounting", "Financial Literacy 104: Want Growth?", "Financial Literacy 105: Strategy Meets Execution"
PS: Selected older articles (2016-18): Blockchain/Crypto/Digital Tokens, Cashless Finance: UPI, Blockchain/Agri & Supply Chain Finance, Digital Agri: Commodity Spot / Futures Markets, Digital Agri: Future of Insurance, From Barter to Blockchain: Brief Journey of Payments & Ledgers, Cashless Financial Inclusion...
LinkedIn: Shivkumar Kalyanaraman
Disclaimer: The views expressed here are personal and meant for information education purposes only and not representative of his employer directly or indirectly.
Twitter: @shivkuma_k
If you like this article, please check out these articles: "Financial & Strategic Value Creation for Techies 101", "Financial & Strategic Value Creation for Techies 102: Analysis & Valuation", "Financial & Strategic Value Creation for Techies 103: Power of Compounding & Discounting", "Electric meets Autonomous", "Commercial Electric Vehicles (EV) Fleets: The Stealth Growth", "Towards Affordable, Ubiquitous, Ultra-Fast EV Charging: Part 1: Need & Battery Issues", "EV Taxi Fleets & Ride Sharing: Poised for Huge Growth", "Shared EV Transportation in India", "Understanding the Rs. 3/kWh bids in India in 2017", "Distributed / Rooftop Solar in India: A Gentle Introduction: Part 1","Rooftop Solar in India: Part 2 {Shadowing, Soiling, Diesel Offset}", "Rooftop Solar in India: Part 3: Policy Tools... Net Metering etc..." "Solar Economics 101: Introduction to LCOE and Grid Parity" , "Solar will get cheaper than coal power much faster than you think..", "Understanding Recent Solar Tariffs in India", "How Electric Scooters,... can spur adoption of Distributed Solar in India," "Solar + Ola! = Sola! ... The Coming Energy-Transportation Nexus in India", "UDAY: Quietly Disentangling India's Power Distribution Sector", "Understanding Solar Finance in India: Part 1", "Back to the Future: The Coming Internet of Energy Networks...", "Tesla Model 3: More than Yet-Another-Car: Ushering in the Energy-Transportation Nexus", "Understanding Solar Finance in India: Part 2 (Project Finance)", "Ola! e-Rickshaws: the dawn of electric mobility in India", "Understanding Solar Finance in India: Part 3 (Solar Business Models)" , "Meet Olli: Fusion of Autonomous Electric Transport, Watson IoT and 3D Printing".